Classification of Lipschitz Derivatives in Terms of Semicontinuity and the Baire Limit Functions

We introduce the generalized notion of semicontinuity of a function defined on a topological space and derive the useful classification of the so-called Lipschitz derivatives of functions defined on a metric space. Secondly, we investigate some connections of the Lipschitz derivatives defined on normed spaces to the Fréchet derivative and relations between little, big and local Lipschitz derivatives (denoted by lip f, 𝕃ip f and Lip f respectively) in terms of Baire limit functions. In particular, we prove that lip f is ℱ_σ-lower, Lip f is ℱ_σ-upper, 𝕃ip f is upper semicontinuous. Moreover, for a function f defined on an open or convex subset of a normed space, the upper Baire limit function of functions lip f and Lip f are equal to 𝕃ip f.